While seemingly simple to the casual observer, the human knee articulates along a complex path. As the knee is flexed, the tibia obviously rotates (flexes) about a coronal axis relative to the femur. However, the femur also translates posteriorly on the tibia and the tibia also rotates about its longitudinal axis. Further, as the knee is flexed, the patella is drawn medially. The complex articulation path of the human knee is dictated primarily by the geometry of the distal femur and proximal tibia. For example, the medial femoral condyle is shorter and spherical in shape, while the lateral femoral condyle is longer and ellipsoidal in shape. The medial tibial condyle is concave whereas the lateral condyle is convex.
The complex path of articulation of the human knee is also dictated by the arrangements of ligaments surrounding and connecting the distal femur and proximal tibia. The human knee is complemented by two collateral ligaments, one on the lateral side of the joint and the other on the medial side thereof. Each ligament is attached to the tibia and the femur. The attachment points to the femur are approximately on the axis of the arc along which the other end of the tibia moves and the knee flexes. The collateral ligaments provide stability to the knee in varus and valgus stresses.
The human knee further includes two cruciate ligaments in the middle of the knee joint. One cruciate ligament is attached to the posterior margin of the tibia, while the other is attached towards the anterior margin of the tibia. Both ligaments are attached to the femur in the notch between the condyles approximately on the axis of the collateral ligaments. The cruciate ligaments provide stability in the anterior and posterior direction, and also allow the knee to rotate axially, i.e., about its longitudinal axis. Thus, as the knee is flexed, the tibia undergoes internal rotation about its longitudinal axis.
Known total knee replacement prostheses generally consist of a femoral component and a tibial component, which are attached to the resected surfaces of the distal femur and the proximal tibia, respectively, either by pressure fitting or by adhering with polymethyl methacrylate bone cement. Each component includes a pair of condylar surfaces that compliment one another and allow the components to articulate relative to one another. The geometry of the complimenting condylar surfaces determines the complexity of movement and degrees of freedom, namely, whether the components can flex, translates and/or rotate relative to one another. The femoral component also includes a patellar flange, which articulates either with the natural patella or an artificial patellar component. The patellar flange provides the lever arm for the quadriceps muscle.
Known total knee prostheses do not accurately replicate the condylar surfaces of the human knee. For example, the femoral condylar surfaces of known prostheses are generally convex and rounded in the medial-lateral direction and anterior-posterior direction. The radius of curvature in the anterior-posterior direction is larger than the radius of curvature in the medial-lateral direction. Generally, the arc center of the sagittal curvature of the distal and posterior aspects of condyles are centered on the axis joining the medial and lateral epicondyles, so that the tension in the collateral ligaments, which attach to the epicondyles, remains nearly constant in flexion and extension. The tibial surfaces are generally concave and dish-shaped with their major axis aligned in the sagittal plane. The sagittal and coronal radii of the tibial condyles are greater than the sagittal and coronal radii of the femoral condyles, which provides some degree of rotational laxity. Likewise, the patellar flange on the femur is concave and oriented from superior to inferior direction with a radius of coronal curvature greater than that of the dome shaped patella.
The design of many prior art total knee replacement components ignore the complex rotational movements of the natural knee in favor of a simple hinge design, which allows only pivotal rotation about a single horizontal axis. Such simple designs have largely been abandoned because of high loosening rates associated with the high rotational stresses placed on the prosthetic components. Other prior art knee prostheses attempt to more closely mimic the motion path of the natural knee. However, these prostheses do not accurately replicate the natural motion path of the human knee and have other manufacturing and durability limitations.
Therefore, it would be desirable to provide a knee replacement prosthesis, which replicates the motion of the natural knee by allowing femoral translation and tibial rotation as the knee is flexed, and which is easy and inexpensive to manufacture.
Many of the prior art knee replacement prostheses are modeled using simple geometries such as circles, arcs, lines, planes, spheres, and cylinders, which have well defined lengths and radii of curvature. However, the complex motion path of the human knee can not be replicated using simple geometries. Prostheses modeled using simple geometries produce unnatural motion, undue tension and pain in the ligaments, and increased wear and loosening of the prosthetic components. Therefore, higher order geometries are needed to generate the complex motion path of the human knee.
Higher order surfaces such as B-spline or Bezier surfaces are much more versatile in describing three dimensional shapes of complex surfaces. A non-uniform rational B-spline surface, or NURBS, is a biparameter surface defined with spline transformations between parameter space and 3D space. NURBS modeling is used in computer graphics for generating and representing curves and surfaces. For example, NURBS is used in CAD modeling software such as those developed by McNeil and associates (Rhinoceros 3D), and Unigraphics (IDEAS). A diagrammatic representation of a NURBS surface is shown in FIG. 25 labeled prior art.
NURBS surface has two independent variables, u and v, and four dependent variables, x(s,t), y(s,t), z(s,t), and d(s,t), such that
                    S        →            ⁡              (                  u          ,          v                )              =          (                                    x            ⁡                          (                              u                ,                v                            )                                /                      d            ⁡                          (                              u                ,                v                            )                                      ,                                            y              ⁡                              (                                  u                  ,                  v                                )                                      /            2                    ⁢                      (                          u              ,              v                        )                          ,                              z            ⁡                          (                              u                ,                v                            )                                /                      d            ⁡                          (                              u                ,                v                            )                                          )                          S        →            ⁡              (                  u          ,          v                )              =                            ∑                      i            =            0                    n                ⁢                              ∑                          j              =              0                        m                    ⁢                                    w              ij                        ⁢                                          P                →                            ij                        ⁢                                          N                                  i                  ,                  p                                            ⁡                              (                u                )                                      ⁢                                          N                                  j                  ,                  q                                            ⁡                              (                v                )                                                                          ∑                      i            =            0                    n                ⁢                              ∑                          j              =              0                        m                    ⁢                                    w              ij                        ⁢                                                            N                  →                                                  i                  ,                  p                                            ⁡                              (                u                )                                      ⁢                                          N                                  j                  ,                  q                                            ⁡                              (                v                )                                                        where the B-spline shape functions N(u) are defined to be:
                              N                      i            ,            0                          ⁡                  (          u          )                    =              {                                            1                                                                        u                  i                                ≤                u                <                                  u                                      i                    +                    1                                                                                                          0                                                                                                          }              ;                                N                      i            ,            p                          ⁡                  (          u          )                    =                                                  u              -                              u                i                                                                    u                                  i                  +                  p                                            -                              u                i                                              ⁢                                    N                              i                ,                                  p                  -                  1                                                      ⁡                          (              u              )                                      +                                                            u                                  i                  +                  p                  +                  1                                            -              u                                                      u                                  i                  +                  p                  +                  1                                            -                              u                                  i                  +                  1                                                              ⁢                                    N                                                i                  +                  1                                ,                                  p                  -                  1                                                      ⁡                          (              u              )                                            ;  given the knot vector ui=u0; u1; . . . ; um 
NURBS modeling is very useful in manufacturing. The three dimensional shape of NURBS surfaces can be readily altered simply by changing the location of the control points. Furthermore, boundary representation solids can be generated with these surfaces, which are easily manufactured. Additionally, the NURBS data can be input into many computer-controlled manufacturing and production machines to program tool paths. Therefore, it would be desirable to provide a knee replacement prosthesis having high order surface geometries generated using NURBS modeling, which replicate the motion of the natural knee and which can be easily manufactured using programmable manufacturing equipment.